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Hereditary evolution processes under impulsive effects. (English) Zbl 1477.34108

Authors’ abstract: In this note, we deal with a model of population dynamics with memory effects subject to instantaneous external actions. We interpret the model as an impulsive Cauchy problem driven by a semilinear differential equation with functional delay. The existence of delayed impulsive solutions to the Cauchy problem leads to the presence of hereditary impulsive dynamics for the model. Furthermore, using the same procedure we study a nonlinear reaction-diffusion model.

MSC:

34K60 Qualitative investigation and simulation of models involving functional-differential equations
34K30 Functional-differential equations in abstract spaces
34K45 Functional-differential equations with impulses
92D25 Population dynamics (general)
35K57 Reaction-diffusion equations
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[1] Appell, J., Measures of noncompactness, condensing operators and fixed points: an application-oriented survey, Fixed Point Theory, 6, 2, 157-229 (2005) · Zbl 1102.47041
[2] Benedetti, I.; Cardinali, T.; Gabor, G.; Rubbioni, P., Lyapunov pairs in semilinear differential problems with state-dependent impulses, Set Valued Var. Anal., 27, 2, 585-604 (2019) · Zbl 1426.34074 · doi:10.1007/s11228-018-0490-7
[3] Benedetti, I.; Rubbioni, P., Existence of solutions on compact and non-compact intervals for semilinear impulsive differential inclusions with delay, Topol. Methods Nonlinear Anal., 32, 227-245 (2008) · Zbl 1189.34125
[4] Bungardi, S.; Cardinali, T.; Rubbioni, P., Nonlocal semilinear integro-differential inclusions via vectorial measures of noncompactness, Appl. Anal., 96, 15, 2526-2544 (2017) · Zbl 1386.34113 · doi:10.1080/00036811.2016.1227969
[5] Burlicǎ, M-D; Necula, M.; Roşu, D.; Vrabie, II, Delay Differential Evolutions Subjected to Nonlocal Initial Conditions. Monographs and Research Notes in Mathematics (2016), Boca Raton: CRC Press, Boca Raton · Zbl 1348.34001
[6] Cardinali, T.; Portigiani, F.; Rubbioni, P., Local mild solutions and impulsive mild solutions for semilinear Cauchy problems involving lower Scorza-Dragoni multifunctions, Topol. Methods Nonlinear Anal., 32, 247-259 (2008) · Zbl 1193.34123
[7] Cardinali, T.; Rubbioni, P., On the existence of mild solutions of semilinear evolution differential inclusions, J. Math. Anal. Appl., 308, 2, 620-635 (2005) · Zbl 1083.34046 · doi:10.1016/j.jmaa.2004.11.049
[8] Cardinali, T.; Rubbioni, P., Impulsive semilinear differential inclusions: topological structure of the solution set and solutions on non-compact domains, Nonlinear Anal., 69, 1, 73-84 (2008) · Zbl 1147.34045 · doi:10.1016/j.na.2007.05.001
[9] Cardinali, T.; Rubbioni, P., Corrigendum and addendum to “On the existence of mild solutions of semilinear evolution differential inclusions” [J. Math. Anal. Appl. 308, no. 2, 620-635] (2005), J. Math. Anal. Appl., 438, 1, 514-517 (2016) · Zbl 1453.34084 · doi:10.1016/j.jmaa.2016.01.066
[10] Cardinali, T.; Rubbioni, P., The controllability of an impulsive integro-differential process with nonlocal feedback controls, Appl. Math. Comput., 347, 29-39 (2019) · Zbl 1428.45007
[11] Denkowski, Z.; Migórski, S.; Papageorgiou, NS, An Introduction to Nonlinear Analysis: Theory (2003), Boston: Kluwer Academic Publishers, Boston · Zbl 1040.46001 · doi:10.1007/978-1-4419-9158-4
[12] Grudzka, A., Ruszkowski, S.: Structure of the solution set to differential inclusions with impulses at variable times. Electron. J. Differ. Equ. 114, 1-16 (2015) · Zbl 1331.34027
[13] Kamenskii, M.; Obukhovskii, V.; Zecca, P., Condensing Multivalued Maps and Semilinear Differential Inclusions in Banach Spaces, De Gruyter Ser. Nonlinear Analysis Applications (2001), Berlin: Walter de Gruyter, Berlin · Zbl 0988.34001
[14] Kornev, SV; Obukhovskii, VV; Zecca, P., Method of generalized integral guiding functions in the problem of the existence of periodic solutions for functional-differential inclusions, Differ. Equ., 52, 10, 1282-1292 (2016) · Zbl 1371.34108 · doi:10.1134/S0012266116100049
[15] Malaguti, L.; Rubbioni, P., Nonsmooth feedback controls of nonlocal dispersal models, Nonlinearity, 29, 3, 823-850 (2016) · Zbl 1351.92043 · doi:10.1088/0951-7715/29/3/823
[16] O’Regan, D.; Precup, R., Existence criteria for integral equations in Banach spaces, J. Inequal. Appl., 6, 77-97 (2001) · Zbl 0993.45011
[17] Pazy, A., Semigroups of Linear Operators and Applications to Partial Differential Equations, Applied Mathematical Sciences (1983), New York: Springer, New York · Zbl 0516.47023
[18] Zhang, X.; Li, Y., Existence of solutions for delay evolution equations with nonlocal conditions, Open Math., 15, 1, 616-627 (2017) · Zbl 1381.34096 · doi:10.1515/math-2017-0055
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